20.500.12556/DiRROS-19045 Relaxations and exact solutions to Quantum Max Cut via the algebraic structure of swap operators The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance $10^{-7}$) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result. Quantum Max Cut swap operators noncommutative polynomials symmetric group Gröbner bases Quantum Max Cut Swap operatorji nekomutativni polinomi simetrična grupa Gröbnerjeve baze true false true Angleški jezik Slovenski jezik Neznano 2024-06-04 11:52:46 2024-06-04 11:52:47 2024-06-05 03:37:49 0000-00-00 00:00:00 2024 0 0 88 str. [article no.] 1352 Vol. 8 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2024-01-01 512 2521-327X 10.22331/q-2024-05-22-1352 197706499 q-2024-05-22-1352.pdf q-2024-05-22-1352.pdf 1 500D62CC503CC3D66CECC70A59C6B302 545d00d0ccca3cd0eb23abf619078f184f44e50ec56cdbe703cf5545b7e702ba 68c7b618-2258-11ef-9c9e-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=26164 https://quantum-journal.org/papers/q-2024-05-22-1352/# 1 34304233-2258-11ef-9c9e-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=26163 Inštitut za matematiko, fiziko in mehaniko 0 0 0